Set-valued fixed point results with application to Fredholm-type integral inclusion

Abstract

In this manuscript, using the concept of a couple $\left[fh\right]$ called generalized upper-class of type-III and $({\alpha }_{\ast },{\eta }_{\ast }{)}_{\text{β}}$-contraction, multi-valued fixed-point results are established in metric spaces. Our results generalized many results in the current literature. Further our methods can be applicable to the study of consensus problems. For the authenticity of the presented work, example and existence theorem for Fredholm-type integral inclusion are also discussed.

Keywords:

• Fixed point
• $({\alpha }_{\ast }{\eta }_{\ast }{)}_{\text{β}}$-contraction
• generalized upper-class of type-III
• Fredholm-type integral inclusion
• $({\alpha }_{\ast }{\eta }_{\ast }A{)}_{\text{β}}$-integral-type contraction

2010 (MOS) SUBJECT CLASSIFICATIONS:

• 35F25
• 45N05
• 47H10

1. Introduction

Fixed-point study for set-valued (SV) mapping first was put forward by Neuman [1] in game theory. Kakutani [2] initiated a fixed-point theory of SV mapping in finite-dimensional spaces. This concept was extended by Bohnenblust and Karlin [3] to infinite-dimensional (Banach) spaces. Nadler [4] generalized the BCP (Banach contraction principle) by presenting the idea of SV contraction known as NCP (Nadler contraction principle), which asserts that in a complete metric space every contraction mapping $T:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ has a fixed point, where ${\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is collection of non-empty bounded and closed subsets of Ξ. He established fixed-point results in metric space utilizing the iterative method and the notion of the Hausdorff metric.

In 2002, the well-known BCP was extended by Branciari [5] to integral-type contraction. Liu et al. established fixed-point results [6] by extended Branciari [5] fixed-point results with the help of restricted type functions. On the other hand, Akram et al. [7] introduced a novel class of contraction mappings known as A-contraction, which is a proper super-class of Kannan's [8], Bianchini's [9] and Reich's [10] type contractions. Sahaa and Deyb [11] established fixed-point theorems for integral A-type contraction mappings.

Samet et al. [12] announced the idea of α-admissible mappings. Related fixed-point results for α-admissible mappings were considered by Hussain et al. [13]. Salimi et al. [4] improved the concept of $(\alpha ,\psi )$-contractive mappings and α-admissible and studied new fixed-point results for such mappings and provided the main results of Samet et al. [12] and Karapinar and Samet [15] as the corollaries. The concept of α-admissible mapping to MV mapping was extended by Asl et al. [16] and Hussain et al. [17]. Radenovic et al. [18] built-up direct proofs of certain common fixed-point results via weak contractive conditions under different condition of control functions. Chandok et al. [19] studied fixed-point results for ${\alpha }_{\text{β}}$-contractive and admissible mappings using the concept of a couple $\left[fh\right]$ called generalized upper-class of type II.

Now, we give some basics defined in a metric space $(\mathrm{\Xi },\varrho )$ for SV mappings. Define the mapping $H:{\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)\times {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)\to {\mathbf{R}}^{+}$ for ${\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2\in {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ by $H({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)=max\left\{\underset{\varsigma \in {\mathrm{\Xi }}_1}{sup}\varrho (\varsigma ,{\mathrm{\Xi }}_2),\underset{\zeta \in {\mathrm{\Xi }}_1}{sup}\varrho (\zeta ,{\mathrm{\Xi }}_1)\right\}$ where $\varrho (\mathrm{\Xi },{\mathrm{\Xi }}_1)=inf\left\{\varrho \right(\mathrm{\Xi },\zeta ):\zeta \in {\mathrm{\Xi }}_1\}$ and $\delta ({\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2)=sup\left\{\varrho \right(\varsigma ,\zeta ):\varsigma \in {\mathrm{\Xi }}_1,\zeta \in {\mathrm{\Xi }}_2\}.$ Geraghty [20] generalized BCP [21] in the following way.

Theorem 1.1

Assume a complete metric space $(\mathrm{\Xi },\varrho )$. Define a mapping $S_1:\mathrm{\Xi }\to \mathrm{\Xi }$. Assume that ∃$\text{β}:{\mathbf{R}}^{+}\to [0,1)$ function such that, for sequence $\left\{t_q\right\}$ of positive real numbers which is bounded, $\text{β}\left(t_q\right)\to 1$ implies $t_q\to 0$ and $\varrho (S_1\zeta ,S_1\varsigma )\le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\varrho (\zeta ,\varsigma ),$ ∀ $\zeta ,\varsigma \in \mathrm{\Xi }$. Then $S_1$ has a unique fixed point.

Definition 1.2

[19]

The function $h:[0,+\mathrm{\infty }{)}^3\to \mathbf{R}$ is a sub-class of type-II if it is continuous and $\zeta ,\varsigma \ge 1\Rightarrow h(1,1,\vartheta )\le h(\zeta ,\varsigma ,\vartheta ).$

Definition 1.3

[19]

Let a continuous function $f:[0,+\mathrm{\infty }{)}^2\to \mathbf{R}$, then the couple $\left[fh\right]$ is upper-class of type-II , if h be a sub-class of type-II with $0\le s_1\le 1\Rightarrow f(s_1,t_1)\le f(1,t_1)$, and $h(1,1,\vartheta )\le f(s_1,t_1)\Rightarrow \vartheta \le s_1{t}_1$.

Definition 1.4

[19]

Let $(\mathrm{\Xi },\varrho )$ be a metric space. Define a mapping $S_1:\mathrm{\Xi }\to \mathrm{\Xi }$. A non-empty subset Q of Ξ is said to be invariant under $S_1,$ if $S_1\zeta \in Q$, for every $\zeta \in Q$.

Definition 1.5

[19]

Let $S_1:\mathrm{\Xi }\to \mathrm{\Xi }$ be a mapping. Moreover, $Q\ne \mathrm{\varnothing }$ be a subset of Ξ, which is invariant under $S_1$ and $\alpha :Q\times Q\to {\mathbf{R}}^{+}$. Then the mapping $S_1$ is an ${\alpha }_Q$-admissible if $\alpha (\zeta ,\varsigma )\ge 1$ ⇒$\alpha (S_1\zeta ,S_1\varsigma )\ge 1$, ∀$\zeta ,\varsigma \in Q$.

$\begin{array}{rl}& \mathrm{\Phi }=\left\{\kappa :\kappa :{\mathbf{R}}^{+}\to {\mathbf{R}}^{+},\kappa \text{ is Lebesgue-integrable,}\right.\\ & \text{summable for every compact subset of}\\ & \left.{\mathbf{R}}^{+}\text{and}{\int }_0^{{ϵ}_1}\kappa \left(\varsigma \right)d\varsigma >0\text{ foreach}{ϵ}_1>0\right\}.\end{array}$

Definition 1.6

Altering distance function is a function $\psi :{\mathbf{R}}^{+}\to {\mathbf{R}}^{+}$ fulfilled the following conditions:

$(Ad_1)$	ψ is non-decreasing and continuous;
$(Ad_2)$	${\psi }^{-1}\left(0\right)=0$.

We signify by Ψ the set of altering distance functions.

Lemma 1.7

[6]

Let $\kappa \in \mathrm{\Phi }$ and $\{{\xi }_q{\}}_{q\in \mathbf{N}}$ is non-negative sequence with $\underset{q\to \mathrm{\infty }}{lim}{\xi }_q=\xi$. Then $\underset{q\to \mathrm{\infty }}{lim}{\int }_0^{{\xi }_q}\kappa \left(\varsigma \right)\mathrm{d}\varsigma ={\int }_0^{\xi }\kappa \left(\varsigma \right)\mathrm{d}\varsigma .$

Lemma 1.8

[6]

Let $\kappa \in \mathrm{\Phi }$ and $\{{\xi }_q{\}}_{q\in \mathbf{N}}$ is non-negative sequence. Then $\underset{q\to \mathrm{\infty }}{lim}{\int }_0^{{\xi }_q}\kappa \left(\varsigma \right)\mathrm{d}\varsigma =0\iff \underset{q\to \mathrm{\infty }}{lim}{\xi }_q=0.$

A-contractions defined by Akram et al. [7] as the following:

Definition 1.9

Let a non-empty set $\mathbf{A}$ contain of all functions $\gamma :{\mathbf{R}}^3\to \mathbf{R}$ satisfying:

$(AC_1)$	γ is continuous on ${\mathbf{R}}^3$ ( w.r.t the Euclidean metric on ${\mathbf{R}}^3$), here ${\mathbf{R}}^3$ is the set of all triplets of non-negative real numbers;
$(AC_2)$	${\kappa }_1\le k{\kappa }_2$ for some $k\in [0,1)$ whenever ${\kappa }_1\le ({\kappa }_1,{\kappa }_2,{\kappa }_2)$ or ${\kappa }_1\le ({\kappa }_2,{\kappa }_1,{\kappa }_2)$ or ${\kappa }_1\le ({\kappa }_2,{\kappa }_2,{\kappa }_1)$ ∀${\kappa }_1,{\kappa }_2$.

Definition 1.10

[7]

A mapping $S_1$ on a metric space $(\mathrm{\Xi },\varrho )$ is said to satisfy A-contraction, if $\varrho (S_1\zeta ,S_1\varsigma )\le \gamma \left(\varrho \right(\zeta ,\varsigma ),\varrho (\zeta ,S_1\zeta ),\varrho (\varsigma ,S_1\varsigma \left)\right)$ ∀$\zeta ,\varsigma \in \mathrm{\Xi }$, $\gamma \in \mathbf{A}$.

Lemma 1.11

[22]

Let $(\mathrm{\Xi },\varrho )$ be a metric space. For any ${\chi }_1,{\chi }_2\in {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right),$ we have $\varrho (\xi ,{\chi }_2)\le H({\chi }_1,{\chi }_2),\mathrm{\forall }\xi \in {\chi }_1$.

Lemma 1.12

[23]

Assume $(\mathrm{\Xi },\varrho )$ is a metric space. Let $\left\{{\zeta }_q\right\}$ be a sequence in Ξ such that $\varrho ({\zeta }_q,{\zeta }_{q+1})\to 0$ as $q\to \mathrm{\infty }$. If $\left\{{\zeta }_q\right\}$ is not a Cauchy sequence, then ∃${ϵ}_1>0$ and sequences of positive integers $\left\{p\right(s\left)\right\}$ and $\left\{q\right(s\left)\right\}$ such that the sequences $\begin{array}{rl}& \varrho ({\zeta }_{p\left(s\right)},{\zeta }_{q\left(s\right)}),\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)+1}),\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)}),\\ & \varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)+1}),\varrho ({\zeta }_{p\left(s\right)+1},{\zeta }_{q\left(s\right)+1})\end{array}$ tend to ${ϵ}_1$ when $s\to \mathrm{\infty }$.

Definition 1.13

[14]

Let $S_1$ be a mapping on a metric space $(\mathrm{\Xi },\varrho )$ and let $\alpha ,\eta :\mathrm{\Xi }\times \mathrm{\Xi }\to {\mathbf{R}}^{+}$ be two mappings. Mapping $S_1$ is an α-admissible w.r.t η if ∀$\zeta ,\varsigma \in \mathrm{\Xi }$ $\alpha (\zeta ,\varsigma )\ge \eta (\zeta ,\varsigma )\Rightarrow \alpha (S_1\zeta ,S_1\varsigma )\ge \eta (S_1\zeta ,S_1\varsigma ).$

Definition 1.14

[16,17]

Let $S_1:\mathrm{\Xi }\to P\left(\mathrm{\Xi }\right)$ be a close valued multi-valued function and $\alpha ,\eta :\mathrm{\Xi }\times \mathrm{\Xi }\to {\mathbf{R}}^{+}$ be two mappings. Mapping $S_1$ is an ${\alpha }_{\ast }$-admissible w.r.t η if ∀$\zeta ,\varsigma \in \mathrm{\Xi }$ the following condition holds $\alpha (\zeta ,\varsigma )\ge \eta (\zeta ,\varsigma )\Rightarrow {\alpha }_{\ast }(S_1\zeta ,S_1\varsigma )\ge {\eta }_{\ast }(S_1\zeta ,S_1\varsigma ),$ where $\begin{array}{rl}& {\alpha }_{\ast }(S_1\zeta ,S_1\varsigma )=inf\left\{\alpha \right(a_1,b_1):a_1\in S_1\zeta ,b_1\in S_1\varsigma \};\\ & {\eta }_{\ast }(S_1\zeta ,S_1\varsigma )=sup\left\{\eta \right(a_1,b_1):a_1\in S_1\zeta ,b_1\in S_1\varsigma \}.\end{array}$

If we take $\alpha (\zeta ,\varsigma )=1$, then mapping $S_1$ is called α-admissible and it is called sub-admissible if we take $\eta (\zeta ,\varsigma )=1$.

Definition 1.15

[19]

Let $(\mathrm{\Xi },\varrho )$ be a metric space. Q a non-empty subset of Ξ, $S_1:\mathrm{\Xi }\to \mathrm{\Xi }$ and $\alpha :Q\times Q\to {\mathbf{R}}^{+}$. A mapping $S_1$ is said to be ${\alpha }_{\text{β}}$-contractive if ∃$\text{β}:{\mathbf{R}}^{+}\to [0,1)$ with the property that $t_q\to 0$ whenever $\text{β}\left(t_q\right)\to 1,$ ∀$\zeta ,\varsigma \in Q$, the following condition holds: $\begin{array}{rl}& h\left(\alpha \right(\zeta ,S_1\zeta ),\alpha (\varsigma ,S_1\varsigma ),\psi (\varrho (S_1\zeta ,S_1\varsigma )\left)\right)\\ & \le f\left(\text{β}\right(\varrho (\zeta ,\varsigma )),\psi (\varrho (\zeta ,\varsigma )\left)\right),\end{array}$ where couple $\left[fh\right]$ is a upper-class of type-II and $\psi \in \mathrm{\Psi }$.

Motivated from the above results, using the idea of Chandok et al. [19] we establish fixed results for SV mapping via $({\alpha }_{\ast },{\eta }_{\ast }{)}_{\text{β}}$-contraction, $({\alpha }_{\ast },{\eta }_{\ast }{)}_{\text{β}}$ -integral-type, $({\alpha }_{\ast },{\eta }_{\ast },A{)}_{\text{β}}$ -integral-type contraction. It is indicating that in our approach the implication $\text{β}\left({\gamma }_n\right)\to 1$ ⇒${\gamma }_n\to 0$ holds for un-bounded sequences $\left({\gamma }_n\right)$. Therefore, our result generalized the results of Hussain et al. [13] and Chandok et al. [19] for SV mapping. Also generalized and extended the result of Liu et al. [6] for SV mapping. As an application existence theorem for Fredholm-type integral inclusion is also established. For some results on the existence of the solutions of nonlinear integral equations and qualitative behaviours of solutions of nonlinear integro-differential equations, we referee the readers to the papers of Deep et al. [24], Tunç [25], Tunç and Tunç [26–29] and the bibliography of these papers.

2. Fixed-point results

In this section, we present a few definitions and new notations which will be utilized throughout in this work. Set $h\left({\vartheta }_1\right):h\left(\eta \right(\zeta ,\varsigma ),\eta (\zeta ,\varsigma ),{\vartheta }_1)$ throughout the paper.

Definition 2.1

$\alpha ,\eta :\mathrm{\Xi }\times \mathrm{\Xi }\to {\mathbf{R}}^{+}$ be two mappings. $h:[0,+\mathrm{\infty }{)}^3\to \mathbf{R}$ is a function of sub-class of type-III if it is continuous and $\begin{array}{rl}& \alpha (\zeta ,\varsigma )\ge \eta (\zeta ,\varsigma )\Rightarrow h\left({\eta }_{\ast }\right(\zeta ,\varsigma ),{\eta }_{\ast }(\zeta ,\varsigma ),{\vartheta }_1)\\ & \le h\left({\alpha }_{\ast }\right(\zeta ,\varsigma ),{\alpha }_{\ast }(\zeta ,\varsigma ),{\vartheta }_2),\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{\vartheta }_1\le {\vartheta }_2.\end{array}$

Definition 2.2

Let $\eta :\mathrm{\Xi }\times \mathrm{\Xi }\to {\mathbf{R}}^{+}$ and $f:[0,+\mathrm{\infty }{)}^2\to \mathbf{R}$ be a mapping. The couple $\left[fh\right]$ is upper-class of type-III if f is a continuous function, h is a sub-class of type-III with $0\le s_1\le 1\Rightarrow f(s_1,t_1)\le f\left(\eta \right(\zeta ,\varsigma ),t_1),$ and $h\left(\eta \right(\zeta ,\varsigma ),\eta (\zeta ,\varsigma ),\vartheta )\le f(s_1,t_1)\Rightarrow \vartheta \le s_1{t}_1.$

Definition 2.3

Assume $(\mathrm{\Xi },\varrho )$ is a metric space and let $S_1:\mathrm{\Xi }\to \mathrm{\Xi }$ be a mapping. A non-empty subset Q of Ξ is said to be invariant under $S_1$ if $S_1\zeta \subseteq Q$, for every $\zeta \in Q$.

Definition 2.4

Let $\alpha ,\eta :Q\times Q\to {\mathbf{R}}^{+}$ be two functions and $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ be a mapping, Q a non-empty subset of Ξ which is invariant under $S_1$. Mapping $S_1$ is an ${{\alpha }_{\ast }}_Q$-admissible if $\alpha (\zeta ,\varsigma )\ge \eta (\zeta ,\varsigma )$ implies ${\alpha }_{\ast }(S_1\zeta ,S_1\varsigma )\ge {\eta }_{\ast }(S_1\zeta ,S_1\varsigma )$, ∀$\zeta ,\varsigma \in Q$.

Definition 2.5

Assume $(\mathrm{\Xi },\varrho )$ is a metric space, Q is a non-empty subset of Ξ, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ and $\alpha :Q\times Q\to {\mathbf{R}}^{+}$. A mapping $S_1$ is said to be ${\alpha }_{\text{β}}$-contractive if ∃$\text{β}:{\mathbf{R}}^{+}\to [0,1)$ with the property that $t_q\to 0$ whenever $\text{β}\left(t_q\right)\to 1$ as well as ∀$\zeta ,\varsigma \in Q$, the following condition holds: (1) $\begin{array}{rl}& h\left({\alpha }_{\ast }\right(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi (H(S_1\zeta ,S_1\varsigma )\left)\right)\\ & \le f\left(\text{β}\right(\varrho (\zeta ,\varsigma )),\psi (\varrho (\zeta ,\varsigma )\left)\right),\end{array}$(1) here couple $\left[fh\right]$ is a upper-class of type-III and $\psi \in \mathrm{\Psi }$.

Theorem 2.6

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space, Q is a non-empty closed subset of Ξ, mapping $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${{\alpha }_{\ast }}_Q$-admissible and Q is invariant under $S_1$. Further assume that $S_1$ is an ${\alpha }_{\text{β}}$-admissible contractive mapping.

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1)$;

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$, we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )\mathrm{\forall }q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Proof.

Let ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ be such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1).$ Since $S_1$ is an ${\alpha }_{\ast }$-admissible mapping, thus ${\alpha }_{\ast }(S_1{\zeta }_0,S_1{\zeta }_1)\ge {\eta }_{\ast }(S_1{\zeta }_0,S_1{\zeta }_1)$. Therefore from (1(1) $\begin{array}{rl}& h\left({\alpha }_{\ast }\right(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi (H(S_1\zeta ,S_1\varsigma )\left)\right)\\ & \le f\left(\text{β}\right(\varrho (\zeta ,\varsigma )),\psi (\varrho (\zeta ,\varsigma )\left)\right),\end{array}$(1) ), we have $\begin{array}{rl}& h\left({\alpha }_{\ast }\right({\zeta }_0,S_1{\zeta }_0),{\alpha }_{\ast }({\zeta }_1,S_1{\zeta }_1),\psi (H(S_1{\zeta }_0,S_1{\zeta }_1)\left)\right)\\ & \le f\left(\text{β}\right(\varrho ({\zeta }_0,{\zeta }_1)),\psi (\varrho ({\zeta }_0,{\zeta }_1)\left)\right).\end{array}$ If ${\zeta }_0={\zeta }_1$, then we have nothing to prove. Let ${\zeta }_0\ne {\zeta }_1$. If ${\zeta }_1\in S_1{\zeta }_1$, then ${\zeta }_1$ is a fixed point of $S_1$. Let ${\zeta }_1\notin S_1{\zeta }_1$. Then using Lemma 1.11 and property of h, we get $\begin{array}{rl}& 0<h\left(\psi \right(\varrho ({\zeta }_1,S_1{\zeta }_1)\left)\right)\le h\left({\alpha }_{\ast }\right({\zeta }_1,S_1{\zeta }_1),{\alpha }_{\ast }({\zeta }_1,S_1{\zeta }_1),\\ & \psi \left(H\right(S_1{\zeta }_0,S_1{\zeta }_1\left)\right)).\end{array}$ From above two equation, we get $0<h\left(\psi \right(\varrho ({\zeta }_1,S_1{\zeta }_1)\left)\right)\le f\left(\text{β}\right(\varrho ({\zeta }_0,{\zeta }_1)),\psi (\varrho ({\zeta }_0,{\zeta }_1)\left)\right).$ Using upper-class of type-III, it follows that $0<\psi \left(\varrho \right({\zeta }_1,S_1{\zeta }_1\left)\right)\le \text{β}\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right)\psi \left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right).$ From property of ψ and β, we have $0<\varrho ({\zeta }_1,S_1{\zeta }_1)<\varrho ({\zeta }_0,{\zeta }_1).$ Hence ∃${\zeta }_2\in S_1{\zeta }_1$ such that $\begin{array}{rl}0<h\left(\psi \right(\varrho ({\zeta }_1,{\zeta }_2)\left)\right)& \le h\left({\alpha }_{\ast }\right(S_1{\zeta }_0,S_1{\zeta }_1),{\alpha }_{\ast }(S_1{\zeta }_0,S_1{\zeta }_1),\\ & \psi \left(H\right(S_1{\zeta }_0,S_1{\zeta }_1\left)\right))\\ & \le f\left(\text{β}\right(\varrho ({\zeta }_0,{\zeta }_1)),\psi (\varrho ({\zeta }_0,{\zeta }_1)\left)\right).\end{array}$ This relation implies that $0<\psi \left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right)\le \text{β}\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right)\psi \left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right).$ It is clear that ${\zeta }_1\ne {\zeta }_2$ $\alpha ({\zeta }_1,{\zeta }_2)\ge \eta ({\zeta }_1,{\zeta }_2).$ Since $S_1$ is an ${\alpha }_{\ast }$-admissible mapping, thus ${\alpha }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2)\ge {\eta }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2)$ $\psi \left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right)\le \text{β}\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right)\psi \left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right)<\psi \left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right)$ If ${\zeta }_2\in S_1{\zeta }_2$, then ${\zeta }_2$ is a fixed point of $S_1$. Let ${\zeta }_2\notin S_1{\zeta }_2$. Then $\begin{array}{rl}0& <h\left(\psi \right(\varrho ({\zeta }_2,S_1{\zeta }_2)\left)\right)\\ & \le h\left({\alpha }_{\ast }\right(S_1{\zeta }_1,S_1{\zeta }_2),{\alpha }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2),\psi (H(S_1{\zeta }_1,S_1{\zeta }_2)\left)\right)\\ & \le f\left(\text{β}\right(\varrho ({\zeta }_1,{\zeta }_2)),\psi (\varrho ({\zeta }_1,{\zeta }_2)\left)\right).\end{array}$ Hence ∃${\zeta }_3\in S_1{\zeta }_2$ such that $\begin{array}{rl}0& <h\left(\psi \right(\varrho ({\zeta }_2,{\zeta }_3)\left)\right)\\ & \le h\left({\alpha }_{\ast }\right(S_1{\zeta }_1,S_1{\zeta }_2),{\alpha }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2),\psi (H(S_1{\zeta }_1,S_1{\zeta }_2)\left)\right)\\ & \le f\left(\text{β}\right(\varrho ({\zeta }_1,{\zeta }_2)),\psi (\varrho ({\zeta }_1,{\zeta }_2)\left)\right).\end{array}$ This relation implies that $0<\psi \left(\varrho \right({\zeta }_2,{\zeta }_3\left)\right)\le \text{β}\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right)\psi \left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right).$ It is clear that ${\zeta }_2\ne {\zeta }_3$ $\alpha ({\zeta }_2,{\zeta }_3)\ge \eta ({\zeta }_2,{\zeta }_3)$ Thus ${\alpha }_{\ast }(S_1{\zeta }_2,S_1{\zeta }_3)\ge {\eta }_{\ast }(S_1{\zeta }_2,S_1{\zeta }_3)$ and $\psi \left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right)\le \text{β}\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right)\psi \left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right)<\psi \left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right).$ Following the same way, we get a sequence $\left\{{\zeta }_q\right\}$ in Q such that ${\zeta }_q\in S_1{\zeta }_{q-1}$ ${\zeta }_q\ne {\zeta }_{q+1},$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$. Thus ${\alpha }_{\ast }(S_1{\zeta }_q,S_1{\zeta }_{q+1})\ge {\eta }_{\ast }(S_1{\zeta }_q,S_1{\zeta }_{q+1})$ (2) $\begin{array}{rl}\psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)& \le \text{β}\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)\psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)\\ & \le \psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right).\\ \psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)& \le \psi \left(\varrho \right({\zeta }_{q-1},{\zeta }_q\left)\right),\end{array}$(2) Hence, we have $\varrho ({\zeta }_q,{\zeta }_{q+1})\le \varrho ({\zeta }_{q-1},{\zeta }_q),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}q\in \mathbf{N}.$ Therefore, the sequence $\varrho ({\zeta }_q,{\zeta }_{q+1})$ is decreasing so ∃ some r>0, such that $\underset{q\to \mathrm{\infty }}{lim}\varrho ({\zeta }_q,{\zeta }_{q+1})=r.$ Letting $q\to \mathrm{\infty }$ in (2(2) $\begin{array}{rl}\psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)& \le \text{β}\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)\psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)\\ & \le \psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right).\\ \psi \left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right)& \le \psi \left(\varrho \right({\zeta }_{q-1},{\zeta }_q\left)\right),\end{array}$(2) ), we have $\underset{q\to \mathrm{\infty }}{lim}\text{β}\left(\varrho \right({\zeta }_{q-1},{\zeta }_q\left)\right)=1$, and this implies that $\underset{q\to \mathrm{\infty }}{lim}\varrho ({\zeta }_q,{\zeta }_{q+1})=0.$ Presently, we will demonstrate that $\left\{{\zeta }_q\right\}$ is a Cauchy sequence. Assume, to the contrary, that $\left\{{\zeta }_q\right\}$ is not a Cauchy sequence. By Lemma 1.12 ∃${ϵ}_1>0$ for which we can discover sub-sequences $\left\{{\zeta }_{q\left(s\right)}\right\}$ and $\left\{{\zeta }_{p\left(s\right)}\right\}$ of $\left\{{\zeta }_q\right\}$ with $n_s>m_s>s$ such that (3) $\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})=\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)-1},{\zeta }_{p\left(s\right)-1})={ϵ}_1.$(3) Setting $\zeta ={\zeta }_{p\left(s\right)-1}$ and $\varsigma ={\zeta }_{q\left(s\right)-1}$ in (1(1) $\begin{array}{rl}& h\left({\alpha }_{\ast }\right(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi (H(S_1\zeta ,S_1\varsigma )\left)\right)\\ & \le f\left(\text{β}\right(\varrho (\zeta ,\varsigma )),\psi (\varrho (\zeta ,\varsigma )\left)\right),\end{array}$(1) ), we obtain $\begin{array}{rl}& h\left({\alpha }_{\ast }\right({\zeta }_{p\left(s\right)-1},{\zeta }_{p\left(s\right)}),{\alpha }_{\ast }({\zeta }_{q\left(s\right)-1},{\zeta }_{q\left(s\right)-1}),\psi (\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})\left)\right)\\ & \le h\left({\alpha }_{\ast }\right({\zeta }_{p\left(s\right)-1},{\zeta }_{p\left(s\right)}),{\alpha }_{\ast }({\zeta }_{q\left(s\right)-1},{\zeta }_{q\left(s\right)-1}),\\ & \psi \left(H\right(S_1{\zeta }_{q\left(s\right)-1},S_1{\zeta }_{p\left(s\right)-1}\left)\right))\\ & \le f\left(\text{β}\right(\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1})),\psi (\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1})\left)\right)\\ & =\text{β}\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)\psi \left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right),\end{array}$ which implies that (4) $\begin{array}{rl}& \psi \left(\varrho \right({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)}\left)\right)\le \text{β}\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)\\ & \psi \left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right),\\ & \frac{\psi \left(\varrho \right({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)}\left)\right)}{\psi \left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)}\le \text{β}\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)<1.\end{array}$(4) Letting $s\to \mathrm{\infty }$ and using (3(3) $\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})=\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)-1},{\zeta }_{p\left(s\right)-1})={ϵ}_1.$(3) ) and (4(4) $\begin{array}{rl}& \psi \left(\varrho \right({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)}\left)\right)\le \text{β}\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)\\ & \psi \left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right),\\ & \frac{\psi \left(\varrho \right({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)}\left)\right)}{\psi \left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)}\le \text{β}\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)<1.\end{array}$(4) ) we get $\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)-1},{\zeta }_{p\left(s\right)-1})=0,$ which is a contradiction. This demonstrates $\left\{{\zeta }_q\right\}$ is a Cauchy sequence and hence is convergent in the complete set Q. Hence ${\zeta }_q\to \vartheta \in Q$ as $q\to \mathrm{\infty }$.

Next, we guess that condition (b) holds. In this case, if, $\alpha ({\zeta }_q,\vartheta )\ge \eta ({\zeta }_q,\vartheta )$, then $\alpha (S_{1}z,S_1{\zeta }_q)\ge \eta (S_1{\zeta }_q,S_1\vartheta )$. Hence, we obtain $\begin{array}{rl}& h\left(\eta \right({\zeta }_q,S_1{\zeta }_q),\eta (z,S_{1}z),\psi (\varrho ({\zeta }_{q+1},S_1\vartheta )\left)\right)\\ & \le h\left(\alpha \right({\zeta }_q,S_1{\zeta }_q),\alpha (\vartheta ,S_1\vartheta ),\psi (\varrho (S_1{\zeta }_q,S_1\vartheta )\left)\right)\\ & \le f\left(\text{β}\right(\varrho ({\zeta }_q,\vartheta )),\psi (\varrho ({\zeta }_q,\vartheta )\left)\right),\end{array}$ which implies that $\psi \left(\varrho \right({\zeta }_{q+1},S_1\vartheta \left)\right)\le \text{β}\left(\varrho \right({\zeta }_q,\vartheta \left)\right)\psi \left(\varrho \right({\zeta }_q,\vartheta \left)\right).$ Taking $q\to \mathrm{\infty }$ and using the properties of ψ and β, we have $\varrho (S_1\vartheta ,\vartheta )=0$, that is, $\vartheta \in S_1\vartheta .$

The methods can be applicable to the study of consensus problems for details, see [30,31].

The following corollaries can be deduced from Theorem 2.6.

Let $h(\zeta ,\varsigma ,\vartheta )=\zeta \varsigma \vartheta$, $f(\zeta ,\varsigma )=\zeta \varsigma$, and $Q=\mathrm{\Xi },$ then

Corollary 2.7

Let $(\mathrm{\Xi },\varrho )$ is a complete metric space, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${\alpha }_{\ast }$-admissible mapping, such that $\begin{array}{rl}& {\alpha }_{\ast }(\zeta ,S_1\zeta ){\alpha }_{\ast }(\varsigma ,S_1\varsigma )\psi \left(H\right(S_1\zeta ,S_1\varsigma \left)\right)\\ & \le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right).\end{array}$

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$, we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )\mathrm{\forall }q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Consider $h(\zeta ,\varsigma ,\vartheta )=\zeta \varsigma \vartheta$, $f(\zeta ,\varsigma )=\zeta \varsigma$, $\psi \left(t\right)=t$ and $Q=\mathrm{\Xi }$, then

Corollary 2.8

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${\alpha }_{\ast }$-admissible mapping, such that $\begin{array}{rl}& {\alpha }_{\ast }(\zeta ,S_1\zeta ){\alpha }_{\ast }(\varsigma ,S_1\varsigma )H(S_1\zeta ,S_1\varsigma )\\ & \le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right).\end{array}$

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$ , we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )$ ∀$q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Let $h(\zeta ,\varsigma ,\vartheta )=\zeta \varsigma \vartheta$, $f(\zeta ,\varsigma )=\zeta \varsigma$, $Q=\mathrm{\Xi }$, $\psi \left(t\right)=t$, ${\alpha }_{\ast }(\zeta ,\varsigma )=1$ then

Corollary 2.9

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${\eta }_{\ast }$-subadmissible mapping, such that $\psi \left(H\right(S_1\zeta ,S_1\varsigma \left)\right)\le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right).$

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\eta ({\zeta }_0,{\zeta }_1)\le 1;$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\eta ({\zeta }_q,{\zeta }_{q+1})\le 1,$ we have $\eta ({\zeta }_q,\zeta )\le 1\mathrm{\forall }q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Let $h(\zeta ,\varsigma ,\vartheta )=(\vartheta +l{)}^{\zeta \varsigma }$,l>1 $f(\zeta ,\varsigma )=\zeta \varsigma +l$, $\psi \left(t\right)=t$ and $Q=\mathrm{\Xi }$, then

Corollary 2.10

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space., $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${\alpha }_{\ast }$-admissible mapping, such that $\begin{array}{rl}& \left(\psi \right(H(S_1\zeta ,S_1\varsigma ))+l{)}^{{\alpha }_{\ast }(\zeta ,S_1\zeta ){\alpha }_{\ast }(\varsigma ,S_1\varsigma )}\\ & \le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right)+l.\end{array}$

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$, we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )$ ∀$q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Let $h(\zeta ,\varsigma ,\vartheta )={\zeta }^m{\varsigma }^n{\vartheta }^p$, $m,n,p\in \mathbf{N}$, $f(\zeta ,\varsigma )={\zeta }^p{\varsigma }^p$, $\psi \left(t\right)=t$ and $Q=\mathrm{\Xi }$, then

Corollary 2.11

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${\alpha }_{\ast }$-admissible mapping, such that $\begin{array}{rl}& \left({\alpha }_{\ast }\right(\zeta ,S_1\zeta \left){)}^m\right({\alpha }_{\ast }(\varsigma ,S_1\varsigma ){)}^n\left(\psi \right(H(S_1\zeta ,S_1\varsigma )){)}^p\\ & \le \left(\text{β}\right(\varrho (\zeta ,\varsigma )\left)\psi \right(\varrho (\zeta ,\varsigma )){)}^p.\end{array}$

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ is a sequence in Q converging to $\zeta \in Q\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1}),$ we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )\mathrm{\forall }q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Let $h(\zeta ,\varsigma ,\vartheta )=(\zeta \varsigma +l{)}^{\vartheta }$, l>1, $f(\zeta ,\varsigma )=(1+m{)}^{\zeta \varsigma }$, m>1 and $Q=\mathrm{\Xi }$ then

Corollary 2.12

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${\alpha }_{\ast }$-admissible mapping, such that $\begin{array}{rl}& \left({\alpha }_{\ast }\right(\zeta ,S_1\zeta \left){\alpha }_{\ast }\right(\varsigma ,S_1\varsigma )+l{)}^{\psi \left(H\right(S_1\zeta ,S_1\varsigma \left)\right)}\\ & \le (1+m{)}^{\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right)}.\end{array}$

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$, we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )\mathrm{\forall }q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Consider $h(\zeta ,\varsigma ,\vartheta )=\left(\right({\zeta }^m+{\zeta }^n{\varsigma }^p+{\varsigma }^q\left)/3\right){\vartheta }^k$, $m,n,p,q,k\in \mathbf{N}$, $f(\zeta ,\varsigma )={\zeta }^k{\varsigma }^k$ and $Q=\mathrm{\Xi }$, then

Corollary 2.13

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${\alpha }_{\ast }$-admissible mapping, such that $\begin{array}{rl}& \frac{\begin{array}{c}\left(\left({\alpha }_{\ast }\right(\zeta ,S_1\zeta ){)}^m+({\alpha }_{\ast }(\zeta ,S_1\zeta ){)}^n\left({\alpha }_{\ast }\right(\varsigma ,S_1\varsigma ){)}^p\right.\\ +\left.\left(\psi \right(H(S_1\zeta ,S_1\varsigma )){)}^q\right)\end{array}}{3}\\ & \le \left(\text{β}\right(\varrho (\zeta ,\varsigma )\left){)}^k\right(\psi \left(\varrho \right(\zeta ,\varsigma \left)\right){)}^k.\end{array}$

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$, we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )\mathrm{\forall }q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Definition 2.14

Assume $(\mathrm{\Xi },\varrho )$ is a metric space, Q is a non-empty subset of Ξ, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ and $\alpha :Q\times Q\to {\mathbf{R}}^{+}$. A mapping $S_1$ is said to be ${\alpha }_{\text{β}}$-integral-type contractive mapping if ∃$\text{β}:{\mathbf{R}}^{+}\to [0,1)$ with the property that $t_q\to 0$ whenever $\text{β}\left(t_q\right)\to 1$ as well as ∀$\zeta ,\varsigma \in Q$, the following condition holds: (5) $\begin{array}{rl}& h\left({\alpha }_{\ast }(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi \left({\int }_0^{\delta (S_1\zeta ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right),\psi \left({\gamma }_1\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\zeta ,S_1\zeta )}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +{\gamma }_2\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\varsigma ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t\\ & +\left.\left.{\gamma }_3\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\zeta ,\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right),\end{array}$(5) where couple $\left[fh\right]$ is a upper-class of type-III and $\psi \in \mathrm{\Psi }$, $\kappa \in \mathrm{\Phi }$ and ${\gamma }_1,{\gamma }_2,{\gamma }_3:{\mathbf{R}}^{+}\to [0,1)$ are functions with (6) $\begin{array}{rl}& {\gamma }_1\left(t\right)+{\gamma }_2\left(t\right)+{\gamma }_3\left(t\right)<1,\mathrm{\forall }t\in {\mathbf{R}}^{\mathbb{+}},\end{array}$(6) (7) $\begin{array}{rl}& \underset{s\to t}{lim sup}\left[{\gamma }_1\right(s)+{\gamma }_2(s)+{\gamma }_3(s\left)\right]<1\mathrm{\forall }t>0.\end{array}$(7)

Theorem 2.15

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space, Q be a non-empty closed subset of Ξ, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${{\alpha }_{\ast }}_Q$-admissible mapping and Q is invariant under $S_1$. Further assume that $S_1$ is an ${\alpha }_{\text{β}}$-admissible integral-type contractive mapping such that the following conditions hold:

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})\mathrm{\forall }q\in \mathbf{N},$ we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )$ ∀$q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Proof.

Let ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ be such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1).$ Since $S_1$ is an ${\alpha }_{\ast }$-admissible mapping, then ${\alpha }_{\ast }(S_1{\zeta }_0,S_1{\zeta }_1)\ge {\eta }_{\ast }(S_1{\zeta }_0,S_1{\zeta }_1)$. Therefore from (5(5) $\begin{array}{rl}& h\left({\alpha }_{\ast }(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi \left({\int }_0^{\delta (S_1\zeta ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right),\psi \left({\gamma }_1\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\zeta ,S_1\zeta )}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +{\gamma }_2\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\varsigma ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t\\ & +\left.\left.{\gamma }_3\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\zeta ,\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right),\end{array}$(5) ), we have $\begin{array}{rl}& h\left({\alpha }_{\ast }({\zeta }_0,S_1{\zeta }_0),{\alpha }_{\ast }({\zeta }_1,S_1{\zeta }_1),\psi \left({\int }_0^{\delta (S_1{\zeta }_0,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right),\psi \left({\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +\left.\left.{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right).\end{array}$ If ${\zeta }_0={\zeta }_1$, then we have nothing to prove. Let ${\zeta }_0\ne {\zeta }_1$. If ${\zeta }_1\in S_1{\zeta }_1$, then ${\zeta }_1$ is a fixed point of $S_1$. Let ${\zeta }_1\notin S_1{\zeta }_1$. Then using property of h, we have $\begin{array}{rl}0& <h\left(\psi \left({\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le h\left({\alpha }_{\ast }({\zeta }_0,S_1{\zeta }_0),{\alpha }_{\ast }({\zeta }_1,S_1{\zeta }_1),\phantom{{\int }_0^{\delta (S_1{\zeta }_0,S_1{\zeta }_1)}}\right.\\ & \left.\psi \left({\int }_0^{\delta (S_1{\zeta }_0,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right).\end{array}$ From the above two equations, it follows that $\begin{array}{rl}0& <h\left(\psi \left({\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right),\psi \left({\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\\ & +\left.\left.{\gamma }_3\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right).\end{array}$ Using upper-class of type-III , we derive that $\begin{array}{rl}0& <\psi \left({\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_0,{\zeta }_1)\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$ Similarly, using properties of β and ψ, we have $\begin{array}{rl}0& <{\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\le {\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\\ & +{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\\ & +{\gamma }_3\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t.\end{array}$ Using Equation (6(6) $\begin{array}{rl}& {\gamma }_1\left(t\right)+{\gamma }_2\left(t\right)+{\gamma }_3\left(t\right)<1,\mathrm{\forall }t\in {\mathbf{R}}^{\mathbb{+}},\end{array}$(6) ) and Lemma 1.11 we have $0<{\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\le {\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t.$ Hence ∃${\zeta }_2\in S_1{\zeta }_1$ such that $\begin{array}{rl}0& <h\left(\psi \left({\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le h\left({\alpha }_{\ast }(S_1{\zeta }_0,S_1{\zeta }_1),{\alpha }_{\ast }(S_1{\zeta }_0,S_1{\zeta }_1),\right.\\ & \left.\psi \left({\int }_0^{\delta (S_1{\zeta }_0,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right),\psi \left({\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\\ & \left.\left.+{\gamma }_3\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right),\end{array}$ which implies that $\begin{array}{rl}0& <\psi \left({\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_0,{\zeta }_1)\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\\ & \left.+{\gamma }_3\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$ It is clear that ${\zeta }_1\ne {\zeta }_2$ $\alpha ({\zeta }_1,{\zeta }_2)\ge \eta ({\zeta }_1,{\zeta }_2),$ Thus $\begin{array}{rl}& {\alpha }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2)\ge {\eta }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2)\\ & \psi \left({\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_0,{\zeta }_1)\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +\left.{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\\ & +\left.\left.\left.{\gamma }_3\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\right)\right)\\ & \le \psi \left({\gamma }_1\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,S_1{\zeta }_0)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +\left.{\gamma }_2\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right)\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_0,{\zeta }_1\left)\right){\int }_0^{\varrho ({\zeta }_0,{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$ If ${\zeta }_2\in S_1{\zeta }_2$, then ${\zeta }_2$ is a fixed point of $S_1$. Let ${\zeta }_2\notin S_1{\zeta }_2$. Then, we have $\begin{array}{rl}0& <h\left(\psi \left({\int }_0^{\varrho ({\zeta }_2,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le h\left({\alpha }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2),{\alpha }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2),\right.\\ & \left.\psi \left({\int }_0^{\delta (S_1{\zeta }_1,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right),\psi \left({\gamma }_1\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_2,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\\ & \left.\left.+{\gamma }_3\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\right).\\ & \le \text{β}\left(\varrho ({\zeta }_1,{\zeta }_2)\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_2,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\\ & \left.\left.+{\gamma }_3\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\right).\end{array}$ Using property of β, ψ and Equation (6(6) $\begin{array}{rl}& {\gamma }_1\left(t\right)+{\gamma }_2\left(t\right)+{\gamma }_3\left(t\right)<1,\mathrm{\forall }t\in {\mathbf{R}}^{\mathbb{+}},\end{array}$(6) ), we get $0<{\int }_0^{\varrho ({\zeta }_2,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\le {\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t.$ Hence, ∃${\zeta }_3\in S_1{\zeta }_2$ such that $\begin{array}{rl}0& <h\left(\psi \left({\int }_0^{\varrho ({\zeta }_2,{\zeta }_3)}\kappa \left(t\right)\mathrm{d}t\right)\right.\\ & \le h\left({\alpha }_{\ast }(S_1{\zeta }_1,S_1{\zeta }_2),{\alpha }_{\ast }\left(S_1{\zeta }_1,S_1{\zeta }_2\right),\phantom{{\int }_0^{\delta (S_1{\zeta }_1,S_1{\zeta }_2)}}\right.\\ & \left.\psi \left({\int }_0^{\delta (S_1{\zeta }_1,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho ({\zeta }_1,{\zeta }_2)\right),\psi \left({\gamma }_1\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_2,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\\ & +\left.\left.{\gamma }_3\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right)\right).\end{array}$ This inequality gives that $\begin{array}{rl}0& <\psi \left({\int }_0^{\varrho ({\zeta }_2,{\zeta }_3)}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_1,{\zeta }_2)\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_2,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$ It is clear that ${\zeta }_2\ne {\zeta }_3\alpha ({\zeta }_2,{\zeta }_3)\ge \eta ({\zeta }_2,{\zeta }_3)$ Thus, we obtain $\begin{array}{rl}& {\alpha }_{\ast }(S_1{\zeta }_2,S_1{\zeta }_3)\ge {\eta }_{\ast }(S_1{\zeta }_2,S_1{\zeta }_3)\\ & 0<\psi \left({\int }_0^{\varrho ({\zeta }_2,{\zeta }_3)}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_1,{\zeta }_2)\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,S_1{\zeta }_1)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_2,S_1{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_1,{\zeta }_2\left)\right){\int }_0^{\varrho ({\zeta }_1,{\zeta }_2)}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$ Continuing this procedure in last, we get a sequence $\left\{{\zeta }_q\right\}$ in Q such that ${\zeta }_q\in S_1{\zeta }_{q-1}$ ${\zeta }_q\ne {\zeta }_{q+1},$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$. Thus, it follows that ${\alpha }_{\ast }(S_1{\zeta }_q,S_1{\zeta }_{q+1})\ge {\eta }_{\ast }(S_1{\zeta }_q,S_1{\zeta }_{q+1})$ $\begin{array}{rl}& \psi \left({\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_n,{\zeta }_{q+1})\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right){\int }_0^{\varrho ({\zeta }_q,S_1{\zeta }_q)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right){\int }_0^{\varrho ({\zeta }_{q+1},S_1{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right){\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \psi \left({\gamma }_1\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right){\int }_0^{\varrho ({\zeta }_q,S_1{\zeta }_q)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right){\int }_0^{\varrho ({\zeta }_{q+1},S_1{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_q,{\zeta }_{q+1}\left)\right){\int }_0^{\varrho ({\zeta }_n,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$ (8) $\begin{array}{rl}& \psi \left({\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_q,{\zeta }_{q+1})\right)\psi \left({\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \psi \left({\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$(8) Using property of ψ and β, we get ${\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\le {\int }_0^{\left(\varrho \right({\zeta }_{q-1},{\zeta }_q)}\kappa \left(t\right)\mathrm{d}t.$ Therefore, the sequence ${\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t$ is decreasing so ∃ some r>0, such that $\underset{q\to \mathrm{\infty }}{lim}{\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t=r.$ Letting $q\to \mathrm{\infty }$ in (8(8) $\begin{array}{rl}& \psi \left({\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_q,{\zeta }_{q+1})\right)\psi \left({\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \psi \left({\int }_0^{\varrho ({\zeta }_q,{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$(8) ), we have $\underset{q\to \mathrm{\infty }}{lim}\text{β}\left(\varrho \right({\zeta }_{q-1},{\zeta }_q\left)\right)=1$, and this implies that $\underset{q\to \mathrm{\infty }}{lim}\varrho ({\zeta }_q,{\zeta }_{q+1})=0.$ Presently, we will demonstrate that $\left\{{\zeta }_q\right\}$ is a Cauchy sequence. Assume, to the contrary, that $\left\{{\zeta }_q\right\}$ is not a Cauchy sequence. By Lemma 1.12 ∃${ϵ}_1>0$ for which we can discover sub-sequences $\left\{{\zeta }_{q\left(s\right)}\right\}$ and $\left\{{\zeta }_{p\left(s\right)}\right\}$ of $\left\{{\zeta }_q\right\}$ with $n_s>m_s>s$ such that (9) $\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})=\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)-1},{\zeta }_{p\left(s\right)-1})={ϵ}_1.$(9) Setting $\zeta ={\zeta }_{p\left(s\right)-1}$ and $\varsigma ={\zeta }_{q\left(s\right)-1}$ in (5(5) $\begin{array}{rl}& h\left({\alpha }_{\ast }(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi \left({\int }_0^{\delta (S_1\zeta ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right),\psi \left({\gamma }_1\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\zeta ,S_1\zeta )}\kappa \left(t\right)\mathrm{d}t\right.\right.\\ & +{\gamma }_2\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\varsigma ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t\\ & +\left.\left.{\gamma }_3\left(\varrho \right(\zeta ,\varsigma \left)\right){\int }_0^{\varrho (\zeta ,\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right),\end{array}$(5) ), we obtain $\begin{array}{rl}& h\left({\alpha }_{\ast }\right({\zeta }_{p\left(s\right)-1},S_1{\zeta }_{p\left(s\right)-1}),{\alpha }_{\ast }({\zeta }_{q\left(s\right)-1},S_1{\zeta }_{q\left(s\right)-1}),\\ & \psi \left({\int }_0^{\delta \left(S_1{\zeta }_{q\left(s\right)-1},S_1{\zeta }_{p\left(s\right)-1}\right)}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le f\left(\text{β}\left(\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1})\right),\right.\\ & \psi \left({\gamma }_1\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},S_1{\zeta }_{p\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{q\left(s\right)},S_1{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\\ & +\left.\left.{\gamma }_3\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le \text{β}\left(\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1})\right)\\ & \psi \left({\gamma }_1\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},S_1{\zeta }_{p\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{q\left(s\right)},S_1{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\\ & +{\gamma }_3\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t,\end{array}$ which implies (10) $\begin{array}{rl}& \psi \left({\int }_0^{\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1})\right)\\ & \psi \left({\gamma }_1\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},S_1{\zeta }_{p\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{q\left(s\right)},S_1{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right),\\ & \frac{\psi \left({\int }_0^{\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})}\kappa \right(t\left)\mathrm{d}t\right)}{\psi \left({\int }_0^{\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})}\kappa \right(t\left)\mathrm{d}t\right)}\le \text{β}\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)<1.\end{array}$(10) Letting $s\to \mathrm{\infty }$ and utilizing (9(9) $\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})=\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)-1},{\zeta }_{p\left(s\right)-1})={ϵ}_1.$(9) ) and (10(10) $\begin{array}{rl}& \psi \left({\int }_0^{\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1})\right)\\ & \psi \left({\gamma }_1\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},S_1{\zeta }_{p\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{q\left(s\right)},S_1{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right){\int }_0^{\varrho ({\zeta }_{p\left(s\right)},{\zeta }_{q\left(s\right)})}\kappa \left(t\right)\mathrm{d}t\right),\\ & \frac{\psi \left({\int }_0^{\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})}\kappa \right(t\left)\mathrm{d}t\right)}{\psi \left({\int }_0^{\varrho ({\zeta }_{q\left(s\right)},{\zeta }_{p\left(s\right)})}\kappa \right(t\left)\mathrm{d}t\right)}\le \text{β}\left(\varrho \right({\zeta }_{p\left(s\right)-1},{\zeta }_{q\left(s\right)-1}\left)\right)<1.\end{array}$(10) ) we get $\underset{s\to \mathrm{\infty }}{lim}\varrho ({\zeta }_{q\left(s\right)-1},{\zeta }_{p\left(s\right)-1})=0,$ which is a contradiction. This exhibits $\left\{{\zeta }_q\right\}$ is a Cauchy sequence and henceforth is convergent in the complete set Q. Thus ${\zeta }_q\to \vartheta \in Q$ as $q\to \mathrm{\infty }$.

Next, we Assume that condition (b) holds. In this manner, $\alpha ({\zeta }_q,\vartheta )\ge \eta ({\zeta }_q,\vartheta )$ then $\alpha (S_1\vartheta ,S_1{\zeta }_q)\ge \eta (S_1{\zeta }_q,S_1\vartheta ),$ we have $\begin{array}{rl}& h\left(\psi \left({\int }_0^{\varrho ({\zeta }_{q+1},S_1\vartheta )}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le h\left(\alpha ({\zeta }_q,S_1{\zeta }_q),\alpha (\vartheta ,S_1\vartheta ),\phantom{{\int }_0^{\varrho ({\zeta }_{q+1},S_{1}z)}}\right.\\ & \left.\psi \left({\int }_0^{\varrho ({\zeta }_{q+1},S_{1}z)}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho ({\zeta }_q,\vartheta )\right),\psi \left(\varrho ({\zeta }_q,\vartheta )\right)\right),\end{array}$ which implies that $\begin{array}{rl}& \psi \left({\int }_0^{\varrho ({\zeta }_{q+1},S_1\vartheta )}\kappa \left(t\right)\mathrm{d}t\right)\\ & \le \text{β}\left(\varrho ({\zeta }_q,\vartheta )\right)\psi \left({\gamma }_1\left(\varrho \right({\zeta }_q,\vartheta \left)\right){\int }_0^{\varrho ({\zeta }_q,S_1{\zeta }_q)}\kappa \left(t\right)\mathrm{d}t\right.\\ & +{\gamma }_2\left(\varrho \right({\zeta }_q,\vartheta \left)\right){\int }_0^{\varrho ({\zeta }_{q+1},S_1{\zeta }_{q+1})}\kappa \left(t\right)\mathrm{d}t\\ & +\left.{\gamma }_3\left(\varrho \right({\zeta }_q,\vartheta \left)\right){\int }_0^{\varrho ({\zeta }_q,\vartheta )}\kappa \left(t\right)\mathrm{d}t\right).\end{array}$ Taking $q\to \mathrm{\infty }$ and using the properties of ψ and β, we have $d(S_1\vartheta ,\vartheta )=0$, that is, $\vartheta \in S_1\vartheta .$

Definition 2.16

Assume $(\mathrm{\Xi },\varrho )$ is a metric space. Q is a non-empty subset of Ξ, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ and $\alpha :Q\times Q\to {\mathbf{R}}^{+}$ are mappings. A mapping $S_1$ is said to be $({\alpha }_{\ast },{\eta }_{\ast },A{)}_{\text{β}}$-admissible integral-type contractive if ∃$\text{β}:{\mathbf{R}}^{+}\to [0,1)$ with the property that $t_q\to 0$ whenever $\text{β}\left(t_q\right)\to 1$ as well as ∀$\zeta ,\varsigma \in Q$, the following condition holds: $\begin{array}{rl}& h\left({\alpha }_{\ast }(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi \left({\int }_0^{H(S_1\zeta ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right)\\ & \le f\left(\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right),\psi \left(\gamma \left({\int }_0^{\varrho (\zeta ,S_1\zeta )}\kappa \left(t\right)\mathrm{d}t,\right.\right.\right.\\ & \left.\left.\left.{\int }_0^{\varrho (\varsigma ,S_1\varsigma )}\kappa \left(t\right)\mathrm{d}t,{\int }_0^{\varrho (\zeta ,\varsigma )}\kappa \left(t\right)\mathrm{d}t\right)\right)\right),\end{array}$ where $\gamma \in \mathbf{A}$ and the couple $\left[fh\right]$ is a upper-class of type-III and $\psi \in \mathrm{\Psi }$.

Similarly, we can prove the following theorem for $({\alpha }_{\ast },{\eta }_{\ast },A{)}_{\text{β}}$ contraction.

Theorem 2.17

Assume $(\mathrm{\Xi },\varrho )$ is a complete metric space. Q is a non-empty closed subset of Ξ, $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ is an ${{\alpha }_{\ast }}_Q$-admissible mapping and Q is invariant under $S_1$. Further assume that $S_1$ is an $({\alpha }_{\ast },{\eta }_{\ast },A{)}_{\text{β}}$-admissible integral-type contractive mapping.

1. There exists ${\zeta }_0\in Q$ and ${\zeta }_1\in S_1{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$

2. For a sequence $\left\{{\zeta }_q\right\}$ in Q converging to $\zeta \in Q$ $\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1})$ ∀ $q\in \mathbf{N},$ we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )$ ∀ $q\in \mathbf{N}$. Then $S_1$ has a fixed point.

Example 2.18

Let $\mathrm{\Xi }=[2,\mathrm{\infty })$ and define a mapping $\varrho :\mathrm{\Xi }\times \mathrm{\Xi }\to {\mathbf{R}}^{+}$ by, $\varrho (\zeta ,\varsigma )=|\zeta -\varsigma |.$ Then $(\mathrm{\Xi },\varrho )$ is a complete metric space. Let $S_1:\mathrm{\Xi }\to {\mathbf{C}}_{\mathbf{b}}\left(\mathrm{\Xi }\right)$ and $\psi :{\mathbf{R}}^{+}\to {\mathbf{R}}^{\mathbb{+}}$, $\alpha ,\eta :\mathrm{\Xi }\times \mathrm{\Xi }\to {\mathbf{R}}^{+}$ defined by, $\begin{array}{rl}{S}_1\left(\zeta \right)& =\left\{\begin{array}{ll}\left\{2\right\}& \mathrm{i}\mathrm{f}\zeta \in [2,4]\\ \left\{0\right\}& \mathrm{i}\mathrm{f}\zeta >4.\end{array}\right.\\ \alpha (\zeta ,\varsigma )& =\left\{\begin{array}{ll}3& \mathrm{i}\mathrm{f}\zeta ,\varsigma \in [2,4]\\ 0& \mathrm{i}\mathrm{f}\zeta ,\varsigma \in {\mathbf{R}}^{+}\setminus [2,4].\end{array}\right.\\ \eta (\zeta ,\varsigma )& =\left\{\begin{array}{ll}1& \mathrm{i}\mathrm{f}\zeta ,\varsigma \in [2,4]\\ 0& \mathrm{i}\mathrm{f}\zeta ,\varsigma \in {\mathbf{R}}^{+}\setminus [2,4].\end{array}\right.\\ \psi \left(t\right)& =t^2.\end{array}$ If $\zeta ,\varsigma \in [2,4],$ then by definitions of ${\alpha }_{\ast }$ and ${\eta }_{\ast }$ we have ${\alpha }_{\ast }(S_1\zeta ,S_1\varsigma )={\alpha }_{\ast }\left(\right\{2\},\{2\left\}\right)=3$ and ${\eta }_{\ast }(S_1\zeta ,S_1\varsigma )={\eta }_{\ast }\left(\right\{2\},\{2\}=1.$ Therefore, it follows that $\alpha (\zeta ,\varsigma )\ge \eta (\zeta ,\varsigma )\Rightarrow {\alpha }_{\ast }(S_1\zeta ,S_1\varsigma )\ge {\eta }_{\ast }(S_1\zeta ,S_1\varsigma ).$ Clearly in other cases, it follows that $\alpha (\zeta ,\varsigma )\ge \eta (\zeta ,\varsigma )\Rightarrow {\alpha }_{\ast }(S_1\zeta ,S_1\varsigma )\ge {\eta }_{\ast }(S_1\zeta ,S_1\varsigma ).$ Hence, $S_1$ is an ${\alpha }_{\ast }$-admissible function w.r.t $\eta \mathrm{\forall }\zeta ,\varsigma \in \mathrm{\Xi }$

For all $\zeta ,\varsigma \in \mathrm{\Xi }$ we discuss the following cases:

(1) If $\zeta ,\varsigma \in [2,4]$, then $\begin{array}{rl}& \left({\alpha }_{\ast }\right(\zeta ,S_1\zeta \left){)}^2\right({\alpha }_{\ast }(\varsigma ,S_1\varsigma ){)}^2\left(\psi \right(H(S_1\zeta ,S_1\varsigma )){)}^2\\ & =\left({\alpha }_{\ast }\right([2,4],3\left){)}^2\right({\alpha }_{\ast }\left(\right[2,4],3){)}^2\left(H\right(\left\{2\right\},\left\{2\right\}){)}^4\\ & =0\le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right).\end{array}$ If $\zeta \in [2,4]$ and $\varsigma >4,$ then $\begin{array}{rl}& \left({\alpha }_{\ast }\right(\zeta ,S_1\zeta \left){)}^2\right({\alpha }_{\ast }(\varsigma ,S_1\varsigma ){)}^2\left(\psi \right(H(S_1\zeta ,S_1\varsigma ){)}^2\\ & =\left({\alpha }_{\ast }\right([2,4],\left\{2\right\}\left){)}^2\right({\alpha }_{\ast }\left(\right(2,\mathrm{\infty }),\{0\left\}\right){)}^2\left(H\right(\left\{2\right\},\left\{0\right\}){)}^4\\ & =0\le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right).\end{array}$ If $\varsigma \in [2,4]$ and $\zeta >4$, then $\begin{array}{rl}& \left({\alpha }_{\ast }\right(\zeta ,S_1\zeta \left){)}^2\right({\alpha }_{\ast }(\varsigma ,S_1\varsigma ){)}^2\left(\psi \right(H(S_1\zeta ,S_1\varsigma )){)}^2\\ & =\left({\alpha }_{\ast }\right((4,\mathrm{\infty }),\left\{0\right\}\left){)}^2\right({\alpha }_{\ast }\left(\right[2,4],\{2\left\}\right)\left){)}^2\right(H\left(\right\{0\},\{2\left\}\right){)}^4\\ & =0\le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\psi \left(\varrho \right(\zeta ,\varsigma \left)\right).\end{array}$ The case $\zeta >4$ and $\varsigma >4$ is easy to calculate. So in all cases by defining $h(\zeta ,\varsigma ,\vartheta )={\zeta }^2{\varsigma }^2{\vartheta }^2,$ and $f(s,t)=st,$ we have $\begin{array}{rl}& h\left({\alpha }_{\ast }\right(\zeta ,S_1\zeta ),{\alpha }_{\ast }(\varsigma ,S_1\varsigma ),\psi (H(S_1\zeta ,S_1\varsigma )\left)\right)\\ & \le f\left(\text{β}\right(\varrho (\zeta ,\varsigma )),\psi (\varrho (\zeta ,\varsigma )\left)\right).\end{array}$ Thus, $S_1$ has a fixed point.

3. Application

SV fixed-point results are explored extensively and have interesting applications in integral and differential inclusion [32–38]. In this section, we derive sufficient conditions for the solutions of Fredholm-type integral inclusion. For this let $\mathrm{\Xi }=C[\kappa ,\upsilon ]$ be the set of all continuous function defined on $[\kappa ,\upsilon ]$. Define $\varrho :\mathrm{\Xi }\times \mathrm{\Xi }\to {\mathbf{R}}^{+}$, by $\varrho (\zeta ,\varsigma )=\underset{t_1\in [\kappa ,\upsilon ]}{sup}|\zeta -\varsigma |,$ is a complete metric space on Ξ. Consider the integral inclusion (11) $\zeta \left(t_1\right)\in \phi \left(t_1\right)+{\int }_{\kappa }^{\upsilon }W(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1,t_1\in [\kappa ,\upsilon ],$(11) where $W:[\kappa ,\upsilon ]\times [\kappa ,\upsilon ]\times \mathbf{R}\to P_{\mathrm{c}\mathrm{v}}\left(\mathbf{R}\right)$ where $P_{\mathrm{c}\mathrm{v}}\left(\mathbf{R}\right)$ is the class of non-empty compact and convex subsets of $\mathbf{R}$. For each $\zeta \in C\left(\right[\kappa ,\upsilon ],\mathbf{R})$ the operator $W(.,.\zeta )$ is lower-semi continuous. Further the function $\phi :[\kappa ,\upsilon ]\to \mathbf{R}$ is continuous.

Define $T:C\left(\right[\kappa ,\upsilon ],\mathbf{R})\to {\mathbf{C}}_{\mathbf{b}}\left(C\right[\kappa ,\upsilon ],\mathbf{R})$ by $\begin{array}{rl}T\zeta \left(t_1\right)& =\{u\in C([\kappa ,\upsilon ],\mathbf{R}):u\in \phi (t_1)\\ & +{\int }_{\kappa }^{\upsilon }W(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1\},\end{array}$ $\zeta \in C\left(\right[\kappa ,\upsilon ],\mathbf{R})$, and set $W_{\zeta }:W(t_1,s_1,\zeta (s_1\left)\right),t_1,s_1\in [\kappa ,\upsilon ]$. Now, for $W:[\kappa ,\upsilon ]\times [\kappa ,\upsilon ]\times \mathbf{R}\to P_{\mathrm{c}\mathrm{v}}\left(\mathbf{R}\right)$ by Michael's selection theorem (MST) there exists a continuous operator $m_{\zeta }:[\kappa ,\upsilon ]\times [\kappa ,\upsilon ]\to \mathbf{R}$ with $m_{\zeta }\in W_{\zeta }$ $\phi \left(t_1\right)+{\int }_{\kappa }^{\upsilon }W(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1\in T\zeta \left(t_1\right),$ which implies that $T\zeta \ne \mathrm{\varnothing }.$ Therefore, $T\zeta$ is closed [34].

Theorem 3.1

Assume that the following assumptions holds:

(A1)	There exists a continuous function $Z:[\kappa ,\upsilon ]\times [\kappa ,\upsilon ]\to [0,\mathrm{\infty })$ such that $H\left(W\right(t_1,s_1,\zeta ),W(t_1,s_1,\varsigma \left)\right)\le Z(t_1,s_1)|\zeta -\varsigma |$ for each $t_1,s_1\in [\kappa ,\upsilon ]$;
(A2)	$\underset{t_1\in [t_1,s_1]}{sup}{\int }_{\kappa }^{\upsilon }|Z(t_1,s_1)|\mathrm{d}{s}_1\le \left(\right(\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\left)/\right({\alpha }_{\ast }(\zeta ,T\zeta ).{\alpha }_{\ast }(\varsigma ,T\varsigma )\left)\right)$.
(A3)	T is an ${\alpha }_{\ast }$-admissible w.r.t $\eta \mathrm{\forall }\zeta ,\varsigma \in \mathrm{\Xi }$ $\alpha (\zeta ,\varsigma )\ge \eta (\zeta ,\varsigma )\Rightarrow {\alpha }_{\ast }(T\zeta ,T\varsigma )\ge {\eta }_{\ast }(T\zeta ,T\varsigma ),$
(A4)	there exists ${\zeta }_0\in \mathrm{\Xi }$ and ${\zeta }_1\in T{\zeta }_0$ such that $\alpha ({\zeta }_0,{\zeta }_1)\ge \eta ({\zeta }_0,{\zeta }_1);$
(A5)	For a sequence $\left\{{\zeta }_q\right\}$ in Ξ converging to $\zeta \in \mathrm{\Xi }\alpha ({\zeta }_q,{\zeta }_{q+1})\ge \eta ({\zeta }_q,{\zeta }_{q+1}),$ we have $\alpha ({\zeta }_q,\zeta )\ge \eta ({\zeta }_q,\zeta )\mathrm{\forall }q\in \mathbf{N}$.

Then the integral inclusion (11(11) $\zeta \left(t_1\right)\in \phi \left(t_1\right)+{\int }_{\kappa }^{\upsilon }W(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1,t_1\in [\kappa ,\upsilon ],$(11) ) has a solution.

Proof.

Let $\zeta ,\varsigma \in \mathrm{\Xi }$ be such that $\varsigma \in T\zeta .$ Then we have $m_{\zeta }(t_1,s_1)\in W_{\zeta }(t_1,s_1)$ for $t_1,s_1\in [\kappa ,\upsilon ]$ such that $u\left(t_1\right)=\phi \left(t_1\right)+{\int }_{\kappa }^{\upsilon }{m}_{\zeta }(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1.$ On other side hypothesis $\left(A_1\right)$ ensures that there exists $v(t_1,s_1)\in M_{\varsigma }(t_1,s_1)$ such that $|m_{\zeta }(t_1,s_1)-v(t_1,s_1)|\le Z(t_1,s_1)|\zeta \left(s_1\right)-\varsigma \left(s_1\right)|.$ Consider the SV operator S defined by $\begin{array}{rl}S(t_1,s_1)& =M_{\varsigma }(t_1,s_1)\cap \{w\in \mathbf{R}:|m_{\zeta }(t_1,s_1)-v(t_1,s_1)|\\ & \le Z(t_1,s_1)|\zeta \left(s_1\right)-\varsigma \left(s_1\right)|\}.\end{array}$ Since the operator T is lower-semi continuous, there exists $m_{\varsigma }:[\kappa ,\upsilon ]\times [\kappa ,\upsilon ]\to \mathbf{R}$ such that $m_{\varsigma }(t_1,s_1)\in S(t_1,s_1)$ for each $t_1,s_1\in [\kappa ,\upsilon ],$ thus $\begin{array}{rl}r\left(t_1\right)& =\phi \left(t_1\right)+{\int }_{\kappa }^{\upsilon }{m}_x(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1\in \phi \left(t_1\right)\\ & +{\int }_{\kappa }^{\upsilon }W(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1,t_1\in [\kappa ,\upsilon ].\end{array}$ Now we have $\begin{array}{rl}& \varrho \left(u\right(t_1),r(t_1\left)\right)=\underset{t_1\in [\kappa ,\upsilon ]}{sup}|u\left(t_1\right)-r\left(t_1\right)|\\ & \le \underset{t_1\in [\kappa ,\upsilon ]}{sup}\left({\int }_{\kappa }^{\upsilon }|m_{\zeta }(t_1,s_1,\zeta (s_1\left)\right)\right.\\ & -\left.m_y\phantom{{\int }_{\kappa }^{\upsilon }}(t_1,s_1,\varsigma (s_1\left)\right)|\mathrm{d}{s}_1\right)\\ & \le \underset{t_1\in [\kappa ,\upsilon ]}{sup}\left({\int }_{\kappa }^{\upsilon }Z(t_1,s_1)|\zeta \left(s_1\right)-\varsigma \left(s_1\right)|\mathrm{d}{s}_1\right)\\ & \le \underset{t_1\in [\kappa ,\upsilon ]}{sup}|\zeta \left(t_1\right)-\varsigma \left(t_1\right)|\underset{t_1\in [\kappa ,\upsilon ]}{sup}{\int }_{\kappa }^{\upsilon }Z(t_1,s_1)\mathrm{d}{s}_1\\ & \le \left(\underset{t_1\in [\kappa ,\upsilon ]}{sup}|\zeta \left(t_1\right)-\varsigma \left(t_1\right)|\right)\frac{\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)}{{\alpha }_{\ast }(\zeta ,T\zeta ){\alpha }_{\ast }(\varsigma ,T\varsigma )}\\ & =\frac{\text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)}{{\alpha }_{\ast }(\zeta ,T\zeta ){\alpha }_{\ast }(\varsigma ,T\varsigma )}\varrho (\zeta ,\varsigma ).\end{array}$ Then, it is clear that ${\alpha }_{\ast }(\zeta ,T\zeta ){\alpha }_{\ast }(\varsigma ,T\varsigma )\varrho \left(u\right(t_1),r(t_1\left)\right))\le \text{β}(\varrho (\zeta ,\varsigma )\left)\varrho \right(\zeta ,\varsigma ),$ which implies that ${\alpha }_{\ast }(\zeta ,T\zeta ){\alpha }_{\ast }(\varsigma ,T\varsigma )H(T\zeta ,T\varsigma )\le \text{β}\left(\varrho \right(\zeta ,\varsigma \left)\right)\varrho (\zeta ,\varsigma ).$ By taking $\psi \left(t_1\right)=t_1$ from corollary 2.8 the integral inclusion (11(11) $\zeta \left(t_1\right)\in \phi \left(t_1\right)+{\int }_{\kappa }^{\upsilon }W(t_1,s_1,\zeta (s_1\left)\right)\mathrm{d}{s}_1,t_1\in [\kappa ,\upsilon ],$(11) ) has a solution.

4. Conclusion

We have used the concept of a couple $\left[fh\right]$ called generalized upper-class of type-III and $({\alpha }_{\ast },{\eta }_{\ast }{)}_{\text{β}}$-contraction, SV fixed results are established in metric spaces. Existence theorem for Fredholm-type integral inclusion are also established. Our result generalized the result of Hussain et al. [13] and Chandok et al. [19] for SV mapping. Also generalized and extended the result of Liu et al. [6] for SV mapping.

Acknowledgements

The authors of this paper are thankful to the editorial board and the anonymous reviewers whose comments improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.